System employing dissipative pseudorandom dynamics for communications and measurement

ABSTRACT

A communication and/or measurement system includes a transmitter that modulates a pseudo-random noise signal with a message signal to produce a wideband signal for transmission. A receiver, which demodulates the wideband signal to recover the message signal, includes an &#34;analog&#34; feedback shift register that reproduces the noise signal based on samples of the received signal. The analog feedback shift register (&#34;AFSR&#34;) is a generalization of a linear feedback shift register (&#34;LFSR&#34;). The AFSR is characterized by a function that agrees with the function that characterizes the LFSR, at the points at which that function is defined. Further, the AFSR characterizing function has stable fixed points at these values, i.e., it has a slope of less than one in these regions. Specifically, the AFSR&#39;s function has stable fixed points at integer values and unstable fixed points at half-integer values and, the stable fixed points act as attractors. The AFSR thus produces a sequence that relaxes to the nearest integer-valued sequence. One example of such a function is: ##EQU1## where the α i  &#39;s are the coefficients of the maximum length polynomial used to produce the wide-band noise signal and y n-1  is the state of the (n-i) th  stage of the register. As long as the samples of received signal that are fed to the AFSR fall within the basins of attraction that surround the stable values, the AFSR can accurately determine the expected next state of the shift register. While this can be done explicitly, the AFSR merges the symbol parsing, acquisition, tracking and update rules into a simple governing equation. The AFSR will thus entrain and produce a binary-valued pseudo-random noise signal. In one embodiment the signal y n  is coupled to the register by a coupling circuit that sums (1-ε)y n  and εS n , to produce the update signal y&#39; n  where ε is a coupling factor selected between 0 and 1, and S n  is a sample of the received signal.

FIELD OF THE INVENTION

The invention relates generally to communication and/or measurementsystems and, more particularly, to systems that use spread spectrummodulation to convert relatively narrow-band information, or message,signals to wide-band signals for transmission.

BACKGROUND OF THE INVENTION

Communication and measurement systems often use spread spectrummodulation techniques to modify message signals for transmission inorder to lower the probability of interception, reduce the peak power ofthe transmitted signals, allow greater channel sharing and/or improveinterference rejection. Further, such systems may also use spreadspectrum techniques to produce high resolution timing or ranginginformation, such as, for example, in global positioning systems. Spreadspectrum modulation involves conversion of a relatively narrow-bandmessage signal into a wide-band signal by multiplying it with, forexample, a "pseudo-random" noise signal. In one arrangement, such as thedirect sequence spread spectrum-system described herein, this involvesamplitude modulation of the noise by the message.

Linear feedback shift registers (LFSR's) are typically used to producethe pseudo-random noise. An LFSR consists of N stages connected togetherto pass their contents forward through the register, with certain stagestapped, or connected, into a feedback path. The feedback path combinesthe contents of the tapped stages and feeds the combination back to oneor more of the stages, to update the register.

The LFSR produces a sequence of symbols, for example, binary symbols orbits, that is periodic but appears random in any portion of the sequencethat is shorter than one period. A period is defined as the longestsequence of symbols produced by the LFSR before the sequence repeats.When this pseudo-random series of symbols is modulated by the message,the result is a wide-band signal with a flat power spectrum over oneperiod of the pseudo-random signal.

The period of the pseudo-random signal is determined by the number ofstages in the shift register and by the feedback between the stages. AnLFSR with "N" stages produces a signal with a period of at most 2^(N) -1bits. If the feedback of the LFSR is set up in accordance with anirreducible polynomial over GF(2), also referred to as a maximum lengthpolynomial, the period of the LFSR is equal to this maximum value, 2^(N)-1. The period can thus be made as long as desired by (i) including inthe register a sufficient number of stages and (ii) combining the stagesin accordance with an associated maximum length polynomial.

The pseudo-random noise is produced by first initializing the LFSR, thatis, setting each of the stages of the LFSR to a predetermined state, andthen shifting the LFSR to produce as the output of the last stage of theregister a sequence of bits. These bits are used to produce thepseudo-random noise signal, which may, for example, have signal valuesof -1 and +1 for corresponding binary values of 0 and 1. This signal isthen modulated by the message signal to produce a signal fortransmission.

A receiver demodulates, or despreads, a received version of thetransmitted signal by reversing the modulation process, i.e., combiningthe received signal with a locally generated replica of the noise signalto reconstruct the desired message. To reproduce the noise signal, thereceiver includes an LFSR that is identical to the one in thetransmitter. The LFSR in the receiver must be in the same state as theone in transmitter, and it must also be operated in synchronism with thereceived signal, to produce the desired message. The receiver must thusdetermine both the state of the LFSR and a clock phase from the receivedsignal. To do this the receiver performs cumbersome search andacquisition operations. Once the LFSR is operating in synchronism withthe received signal, the receiver must perform operations thataccurately track the received signal, so that the LFSR continues tooperate in synchronism with the signal.

While the foregoing operations are usually applied to messages indigital, i.e., binary form, they can also be applied to analog orcontinuous-valued bounded messages, for example, messages whoseinstantaneous values lie anywhere in the range -1 to +1.

To ensure that the receiver synchronizes to and remains in synchronismwith the received signal, some prior known systems use chaoticmodulation signals. Synchronization, or entrainment, is ensured in achaotic system that is non-linear, dissipative and in which thetransmitter and the receiver are coupled such that their joint Lyapunovexponent is negative.

In such a communication system the transmitter generates a chaotic noisesignal and modulates this signal by the message signal to produce achaotic signal for transmission. A receiver in the chaotic systemmanipulates the transmitted signal, by applying that signal to its ownchaotic noise signal generator, thereby synchronizing this generator tothe one in the transmitter and recovering the message. An example ofsuch a system is discussed in U.S. Pat. No. 5,291,555 to Cuomo et al.

The noise signal produced by a chaotic system is randomly driven,because of the exponential amplification of small fluctuations. Thesystem is not an ideal noise source, however. There can be linearcorrelations in the signal that lead to undesirable peaks in the powerspectrum that must be filtered for optimum use of available bandwidth.Further, even after filtering to flatten the power spectrum there remainnon-linear correlations, which can interfere with subsequent coders ormake the system more susceptible to unintended reception.

The filters required to flatten these peaks are at best complex, and maynot be realistically or economically feasible. Also, the filtersrequired in the receiver to restore the peaks are also complex and maybe infeasible. If so, the transmitter may have to transmit with reducedpower, which may adversely affect the reception of the signal.

Moreover, if the receiver in the chaotic system is to synchronize to thetransmitted signal within a reasonable time, the message signal cannotbe too large when compared to the chaotic carrier signal. If the messagesignal is too small, however, the transmitted signal is comprised mainlyof the chaotic carrier and bandwidth is wasted.

As in any communication system, there is a trade-off between time tosynchronize, or lock, to a received signal and the robustness of thesystem, that is, the accuracy with which the system locks to the signaland remains locked thereafter. Known chaotic systems cannot be readilyaltered to change in a predictable way their attractor dimensions, i.e.,the usable numbers of degrees of freedom. These systems thus cannotreadily alter the trade-off between time to lock and robustness.

SUMMARY OF THE INVENTION

The invention is a system that modulates a pseudo-random noise with ananalog message signal and includes, in a receiver that demodulates themodulated signal, an "analog" generalization of a linear feedback shiftregister. The analog feedback shift register (AFSR), which is bothnon-linear and dissipative, uses directly samples of the received signalto synchronize to that signal. The AFSR is thus coupled to thetransmitter through the received signal. The system is non-chaotic anduses non-correlated (i.e., "ideal") pseudo-random noise to modulate amessage. Synchronization of the transmitter and receiver in thisnonlinear and dissipative system is possible because of the coupling.

In general, the AFSR is characterized by a function A(x) that is ageneralization of the function that adds (modulo-two) the contents ofvarious stages of an analogous linear feedback shift register (LFSR).The map of x_(n+1) =A(x_(n)) has stable (attracting) fixed points atinteger values and unstable (repelling) fixed points at half-integervalues. Consequently, for real-valued initial conditions, the sequenceproduced from the function A(x) relaxes on to the "nearest"binary-valued maximal sequence of 1s and 0s of the corresponding LFSR.This is discussed in more detail in Section B below.

The AFSR in the receiver uses a function A_(r) (x) that is derived fromthe function that governs the operation of the associated LFSR, that is,the register used in the transmitter to modify the message signal. TheAFSR's characterizing equation thus has stable limit cycles identical tothe binary values, i.e., at 1 and 0. These limit cycles are "attractors"and the AFSR drives any values that are within "basins of attraction"associated with these attractors to the stable values. The AFSR thusproduces an essentially binary-valued noise signal from an input signalconsisting of samples of an analog received signal. This eliminatesquantitization errors associated with assigning binary values to thereceived signal during demodulation, as is required with prior knownsystems that use LFSRs in both the transmitter and the receiver.

In an exemplary embodiment, the update expression for an N-stage AFSRis: ##EQU2## where the α_(i) 's are the coefficients of the maximumlength polynomial governing the operation of the associated LFSR,y_(n-i) is the state of the (n-i)^(th) stage of the register, and y_(n)is the feedback signal.

The AFSR also includes coupling circuitry, which combines the receivedsignal with the feedback signal y_(n), to produce an update signal,y'_(n), for the first stage of the register. This coupling circuitrymultiplies a measured sample S_(n) of the received signal, or, in analternative embodiment, a value that represents the sign of the signalsample, by a selected coupling factor, ε, which is between 0 and 1. Itthen adds the product, for example, εS_(n), to the product (1-ε)y_(n)and applies the sum y'_(n) to the first stage of the AFSR. The selectionof a value for ε determines the degree of coupling between thetransmitter and the receiver, which, as discussed in more detail below,determines how quickly synchronization is achieved and how immune theregister is to a corrupted received signal. The operation of thecoupling circuitry is also discussed in more detail below.

In an alternative embodiment of the system, the coupling circuitryupdates the AFSR, at selected times, with the feedback signal y_(n), andat other times with the calculated value y'_(n). In particular, it usesthe feedback signal to update the AFSR when the amplitude of thereceived signal differs by more than a predetermined amount from thebinary signal values. In this embodiment, it is convenient to use binarysignal values of -1 and +1, rather than 0 and 1. The coupling circuitryuses the feedback signal to update the AFSR when |(|S_(n) |-1)|>δ, whereδ is chosen between 0 and 2-μ and μ is a modulation factor. When|(|S_(n) |-1)|>δ, it uses the calculated values y'_(n) =(1-ε)y_(n)+εS_(n). As discussed in more detail below, the system applies to theAFSR signal samples S_(n) that provide good estimates of thepseudorandom noise of the transmitter, and omits the samples that donot. Using this selection technique, the system can obtain and maintainsynchronism with a received signal that includes a relatively largemessage signal.

AFSRs may be used in both the transmitter and the receiver, and in oneconfiguration the two registers may have identical feedback paths. TheAFSR in the receiver may thus couple a portion of the message into thesignal that updates the register. This reduces demodulation errors inthe receiver once lock has been achieved.

The current system combines the best features of the prior known chaoticand non-chaotic systems. The current system is a non-chaotic system thatuses pseudo-random noise to spread the spectrum of the message signal,and thus, has the advantage of a flat power spectrum for the transmittedsignal. The inventive system is also both non-linear and dissipative,which means that the receiver, coupled to the transmitter by thetransmitted signal, can entrain, or synchronize, even when thetransmitted signal contains a message that is relatively large whencompared to the carrier. The signals transmitted using the currentsystem thus include more information in the transmitted signal and wasteless bandwidth than the signals transmitted by prior known chaoticsystems. Further, the current system has the advantage of not requiringthe assignment of binary, or digital, values to the received signalbefore recovery of the message, and thus, avoids synchronizationproblems that may be caused by quantitization errors.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and further advantages of the invention may be betterunderstood by referring to the following description in conjunction withthe accompanying drawings, in which:

FIG. 1 is a functional block diagram of a signal transmission systemconstructed in accordance with the invention;

FIG. 2 is a functional block diagram that depicts in more detail alinear feedback shift register and an exemplary analog feedback shiftregister that are included in the system of FIG. 1;

FIGS. 3A-3C are graphs depicting synchronization time, error and errorvariance as a function of a coupling factor associated with the analogfeedback shift register of FIG. 2; and

FIG. 4 is a diagram that illustrates the theory of the analog feedbackshift register with the dotted line representing an addition modulo twofunction that characterizes a linear feedback shift register and thesolid line representing the periodic function that characterizes theanalog feedback shift register;

FIG. 5 is a graph depicting the operation of an illustrative system inacquiring a signal;

FIG. 6 is a functional block diagram of an analog feedback shiftregister constructed in accordance with a second embodiment; and

FIG. 7 is a functional block diagram of an analog feedback shiftregister implemented as an optical system.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

I first discuss in Section A a system that includes in a receiver an"analog" feedback shift register (AFSR). I then discuss in Section B theoperative theory of the system. In Section C I discuss an alternativeembodiment of the system, and in Section D I discuss an alternativeimplementation of the AFSR.

A. A First Embodiment of the System

FIG. 1 depicts a signal transmission system 10 with a transmitter 12that produces a wide-band signal, t_(n), for transmission. Thetransmitter 12 includes a linear feedback shift register (LFSR) 14 thatoperates in a conventional manner to produce a pseudo-random noisesignal x_(n). A modulator 16 modulates the pseudo-random noise signal byan analog message signal, m_(n), to produce t_(n) =x_(n) (1+μm_(n)),where μ is the modulation factor. The transmitter 12 then transmits themodulated signal t_(n) to a receiver 20 over a communications channel18. The LFSR 14 is discussed in more detail with reference to FIG. 2below.

The receiver 20 receives a version of the transmitted signal that mayinclude channel noise. The receiver applies signals associated withsamples, S_(n), of the received signal to an "analog" feedback shiftregister (AFSR) 22, which, as discussed in more detail with reference toFIG. 2 below, uses non-chaotic, non-linear feedback to reproduce asignal, y_(n), that is a synchronized version of the pseudo-random noisesignal x_(n). A demodulator 24 uses the reproduced noise signal torecover the message signal from the received signal by calculating:##EQU3##

Referring now to FIG. 2, LFSR 14 has N stages 15 and is organized inaccordance with a maximum length polynomial over GF(2). It ischaracterized by the following update expression: ##EQU4## where theα_(i) 's are the coefficients of the maximum length polynomial andx_(n-i) is the state of the (n-i)^(th) stage of the LFSR 14. Modulo-twoadders 26 combine the contents of the associated stages 15 in accordancewith the update expression, to generate a feedback signal, x_(n), thatis applied to update the first stage 15 of the register. The feedbacksignal is also applied to the modulator 16, which multiplies it by amessage signal that is limited by the modulation factor, that is, thesignal μm_(n). In the example, the modulator 16 includes a shifter 17that shifts the message signal to produce a signal 1+μm_(n), and amultiplier 18 that multiplies the shifted signal by the LFSR feedbacksignal, x_(n). The modulator thus produces for transmission themodulated signal t_(n) =x_(n) (1+μm_(n)).

The update equation for the AFSR 22 is: ##EQU5## where the α_(i) 's arethe coefficients of the maximum length polynomial that characterizes theLFSR 14, and y_(n-1) is the state of the (n-i)^(th) stage 30 of theAFSR. As discussed in more detail in Section B, this equation has stablelimit cycles with period 2^(N) -1 at binary integer values. The stablelimit cycles are "attractors", and the AFSR essentially drives anyvalues that are within "basins of attraction" associated with theseattractors to the stable, that is, the binary, limit cycle values.

The feedback path of the AFSR includes floating point processors 31, ortheir analog equivalent. These processors 31 combine the contents of thetapped stages 30 in accordance with the update expression. As discussedabove, the AFSR uses directly the samples of the received signal--anddoes not require that these samples be translated to binary values.

The feedback signal y_(n) produced by the feedback path is coupled tothe first stage 30 through coupling circuit 32. The coupling circuit 32updates the AFSR 22 with an update signal, y'_(n), that is a combinationof the feedback signal, y_(n), and the received signal, S_(n). Acoupling factor, ε, determines how much of the received signal, and inturn, how much of the feedback signal, are included in the updatesignal. The coupling factor is discussed in more detail below.

The coupling circuitry 32 includes floating point processors 34, 36 and38 or their analog equivalents. Multiplier 34 multiplies samples of thereceived signal by the coupling factor ε, multiplier 36 multiplies thefeedback signal y_(n) by 1-ε, and processor 38 then adds the products ofthe multipliers, to produce an update signal:

    y'.sub.n =εs.sub.n +(1-ε)y.sub.n.

The feedback signal y_(n) is also applied to the demodulator 24. Thedemodulator 24 includes a subtractor 25 that subtracts the feedbacksignal from the received signal, and a divider 26 that divides the sumby μ times the feedback signal, to recover the message signal from thereceived signal.

The receiver 20 recovers a clock phase from the received signal in aconventional manner, using, for example, a phased-locked loop (notshown) and monitoring transitions in the received signal. The receiveralso determines that the AFSR 22 and the received signal are insynchronism in a conventional manner.

The coupling factor, ε, is between 0 and 1. When it is closer to 1, moreof the received signal is included in the update signal. This tends toallow the AFSR 22 to synchronize more rapidly to the received signal. Ifthe coupling factor is too close to 1, however, corruption of thereceived signal may adversely affect synchronization.

When the coupling factor ε is closer to 0, more of the feedback signaland less of the received signal to be applied to the register. Theregister may thus take longer to synchronize. It can more easily handlea corrupted received signal, however, since less of the signal is usedto update the register. There is thus a trade-off between the time ittakes for the register to synchronize and the accuracy of thesynchronization.

The coupling factor may be chosen initially closer to 1, for fastsynchronization, and thereafter chosen closer to 0 for enhanced handlingof signal corruption.

FIG. 3A depicts graphically the time it takes to synchronize the AFSR 22as a function of the coupling factor ε. FIG. 3B depicts synchronizationerror, i.e., |x_(n) -y_(n) |, as a function of ε and FIG. 3C depicts thestandard deviation of the synchronization error as a function of ε. Asdiscussed above, the coupling circuitry may set the coupling factor εinitially close to 1 to achieve faster synchronization and thereafter(when, for example, the error rate in the recovered message issufficiently low) it may set ε closer to 0.

The AFSR 22 may be used in any system that modulates a message signalwith pseudo-random noise. It may thus be used in systems that simplymodify the message signal by multiplying it by the pseudo-random noiseor in systems that utilize more complex modulation schemes. Further, theAFSR 22 may be included in both the transmitter and the receiver. TheAFSR in the transmitter will necessarily produce the same pseudo-randomnoise signal as the LFSR 14, since the AFSR in the transmitter isupdated by its feedback signal, which is attracted to binary values.

In a system with AFSRs in both the transmitter and the receiver, the tworegisters, and thus, their feedback paths may be identical. The AFSR inthe transmitter couples into the signal fed back to update the registera portion of the message, through coupling circuitry identical to thatin the receiver. This reduces demodulation errors in the receiver, oncethe two AFSRs are operating in synchronism.

B. The Theory

The function L(x)=x_(n) (mod 2) that characterizes the LFSR has forGF(2), i.e., binary systems, the following properties:

    L(x)=0 if x is an even integer

    L(x)=1 if x is an odd integer.

The function is thus defined at integer values. The state space of theLFSR comprises the corners of an N-dimensional hypercube, which has oneaxis for each time delay, or stage, of the register. The sequenceproduced by the LFSR "visits" each corner of the hypercube once during aperiod.

When the function L(x) is generalized to A(x) for the AFSR, therecursive relation x_(n+1) =A(x_(n)) has stable fixed points at integervalues and unstable fixed points at half-integer values. For real-valuedx's, a sequence produced by A(x) will relax onto the "nearest"binary-valued sequence since it is repelled by the unstable half-integervalues and attracted to the stable integer values, as depicted in FIG. 4in which the dotted line represents the addition modulo-two functionL(x) and the solid line represents the periodic AFSR function A(x). Thefunction A(x) has the following properties: ##EQU6## An example of sucha function is the cosine function. It is not necessary that the functionbe symmetrical about the extrema, only that the slope has a magnitude ofless than one about these points.

The AFSR has an attracting basin around the limit cycle of period 2^(N).In state space the corners of the associated N-dimensional hypercubebecome attractors, with basins of attraction around each of themdictated by the AFSR update expression. The AFSR, through its feedbackpath, drives sample values that are within the basins of attraction tothe appropriate corners of the hybercube, and thus, to the stableinteger values. As long as the samples of received signal that are fedto the AFSR fall within these basins of attraction, the AFSR canaccurately determine the expected next state of the shift register. TheAFSR will thus entrain and produce a binary-valued pseudo-random noisesignal. While this can be done explicitly, the AFSR merges the sampleparsing, acquisition, tracking and update rules into a simple governingequation.

C. A Second Embodiment of the System

Referring now to FIG. 6, the AFSR 23 includes a coupling circuit 33 thatselectively couples to the first stage of the AFSR as an update signal(i) a signal that includes information associated with a received signalsample; or (ii) the feedback signal produced by the AFSR. In a system inwhich 0 and 1 binary values are represented by signal values -1 and 1,respectively, the coupling circuit includes a comparator 39 thatcompares the quantity |(|S_(n) |-1)| with a predetermined maximum valueδ, to determine essentially if the received signal sample provides agood estimate of the transmitter pseudo-random signal produced by theLFSR or the AFSR in the transmitter. If the comparator 39 determinesthat the quantity does not exceed the predetermined value, it directs aselector 37 to apply to the first stage of the AFSR as an update signal

    y'.sub.n =εsgnS.sub.n +(1-ε)y.sub.n

where sgnS_(n) is the sign of the signal sample.

If the comparator 39 determines that |(|S_(n) |-1) | is greater than δ,it directs the selector 37 to apply to the first stage of the AFSR thefeedback signal y_(n). The coupling circuitry thus refrains fromincluding the corresponding received signal information in the updatesignal, and prevents a signal sample that provides a bad estimate of thetransmitter's LFSR state [is outside the basin of attraction] fromadversely influencing the state of the AFSR. The value of δ is chosensuch that 0<δ≦2-μ.

In systems that use -1 and 1 as the binary signal values, thecharacterizing equation for the AFSR is ##EQU7##

In this embodiment the use of the "selective" perturbations to the AFSR,i.e., the selective inclusion of information relating to the receivedsignal and the update signal, is similar in spirit to the occasionalfeedback used in controlling conventional chaotic systems. See, forexample, E. Ott et al., Controlling Chaos, Phys. Rev. Lett. Vol. 64, no.11, p. 1196-99 (1990); R. Roy et al., Dynamical Control of A ChaoticLaser: Experimental Stabilization of a Globally Coupled System Phys.Rev. Lett. Vol. 68, no. 9, p. 1259-62 1259 (1992).

In the system described in Section A above there is an implicitrestriction on the size of the modulation factor, μ. If the messagesignal, μm, is relatively large when compared to the pseudo-random noisesignal, samples of the transmitted signal x_(n) (1+μm_(n)) may havevalues that differ significantly from the associated pseudo-random noisesignal. These values may even have different signs, which may makeambiguous a next state of the AFSR, and thus, adversely affectsynchronization. The value of μ also affects the time it takes tosynchronize to the received signal. With a relatively large μ, thesystem tends to require a longer time to lock to the received signal.Using the selective feedback, these adverse effects of using a larger μcan be minimized. As illustrated in FIG. 5, the current system locksrelatively quickly to the signal, even with a μ=1.8.

D. An AFSR Implemented as an Optical System

FIG. 7 depicts a two-tap AFSR 40 that is implemented as an opticalsystem. A pulse-shaping amplifier 44 produces pulses in response to anincoming signal that passes through a controllable phase shifter 42. Thepulses produced by the amplifier 44 are successively applied to beamsplitters 46 and 48 that are spaced in accordance with the associatedpseudorandom code, and are thus spaced apart by an integral number ofcode chips. The beam splitters 46 and 48 are analogous to the delay tapsof the AFSR of FIG. 1.

The light deflected by the beam splitter 46 along path 47 is coherentlycombined in a beam splitter 52 with the light deflected along path 49 bythe beam splitter 48 and then reflected along path 51 by mirror 50. Theresultant beam is steered by a mirror 54 to control the phase shifter42. Specifically, the phase shifter 42, which is constructed ofphoto-refractive material, imparts a phase shift to the incoming signalthat is proportional to the intensity of the light from the mirror 54.In the illustrative embodiment, the beam splitters and mirrors arespaced so that there is no appreciable delay relative to the chip lengthof the code. This ensures that the appropriate pulses are combined inaccordance with the code, and thus, that when the system is synchronizedto the incoming signal that the output signals passed by the beamsplitter 48 are the pseudorandom code signals.

The beam splitters and mirrors are also positioned such that the path 47from the beam splitter 46 to the beam splitter 52 and the combined paths49 and 51 from the beam splitter 48 to the mirror 52 differ by an oddnumber of half-wavelengths, such that the signals along these paths areeither in-phase or 180° out-of-phase. The signals thus cancel or add atbeam splitter 52, which is analogous to the adders in the feedback pathof the AFSR of FIG. 1. The combining of the deflected light and the useof that light to control the phase shifter 42 provides to the system thenon-linearity that is analogous to the cosine function of the AFSR ofFIG. 1. Accordingly, the system has attractors that drive the system toproduce signals that operate the phase shifter 42 in two modes--one inwhich it shifts the phase of the incoming signal by 180° and one inwhich it shifts the phase of the incoming signal by 0°.

The signals from by beam splitter 52 are spread spectrum, or wide band.These signals interfere primarily over only a narrow band centered onthe carrier frequency. A filter 56 filters from the wide band signal thefrequencies that are outside of the narrow band over which the signalsinterfere. An amplifier 58 amplifies the filtered signals to producesignals in which the differences in signal intensities enable the phaseshifter, when the AFSR is in synchronism with the incoming signal, tooperate in the two modes of operation in which it shifts the phase ofthe signal by 180° or by 0°. The rate at which the transmitted signal ismodulated should be selected such that the signal energy in the passband of the filter 56 is sufficient to overcome any noise within thatband.

While the system of FIG. 7 is discussed in terms of using opticalsignals, it may instead be implemented using any coherent radiation,such as microwaves, ultrasound, and so forth.

The foregoing description has been limited to a specific embodiment ofthis invention. It will be apparent, however, that variations andmodifications may be made to the invention, with the attainment of someor all of its advantages. Therefore, it is the object of the appendedclaims to cover all such variations and modifications as come within thetrue spirit and scope of the invention.

What is claimed is:
 1. A communication or measurement systemincluding:A. a transmitter for transmitting a modulated signal, thetransmitter including a first feedback shift register that produces apseudo-random noise signal that is used to modulate a message signal; B.a receiver for receiving the modulated message signal, the receiverincluding a second feedback shift register for reproducing thepseudo-random noise signal from the received modulated signal and usingthe noise signal to recover the message signal from the received signal,the second feedback shift register being characterized by a non-chaoticexpression that is associated with the expression that governs the firstfeedback shift register and has stable fixed points at integer valuesand unstable fixed points at 1/2 integer values.
 2. The communicationsystem of claim 1, wherein the second feedback shift register ischaracterized by a function that has at integer values a slope ofmagnitude less than
 1. 3. The system of claim 1, whereini. the firstfeedback shift register is characterized by the following updateexpression: ##EQU8## where the α_(i) 's are the coefficients of amaximum length polynomial and x_(n-i) is the state of the (n-i)^(th)stage of the register, and ii. the second feedback shift register ischaracterized by the following update expression: ##EQU9## where theα_(i) 's are the coefficients of the maximum length polynomial andy_(n-1) is the state of the (n-i)^(th) stage of the register.
 4. Thesystem of claim 2 further including a coupling circuit that produces anupdate signal, y'_(n), for the second shift register, y'_(n) =(1-ε)y_(n)+εS_(n), where ε is a coupling factor selected between 0 and 1 and S_(n)is a sample of the received signal, the coupling circuit applying theupdate signal to a first stage of the second shift register.
 5. Thesystem of claim 4, wherein the coupling factor is set to a valuerelatively close to 1 until the second shift register synchronizes tothe received signal, and is thereafter decreased.
 6. The system of claim3, further including a coupling circuit that produces a signal y'_(n)=(1-ε)y_(n) +εsgnS_(n), where ε is a coupling factor selected between 0and 1 and sgnS_(n) is the sign of a sample of the received signal, thecoupling circuit updating the second feedback shift register using thesignal y'_(n) if the amplitude of the received signal sample does notexceed a signal value that represents a binary value by more that apredetermined amount.
 7. An analog feedback shift register for use in acommunication or measurement system, the shift register including:A. aplurality of stages connected together such that the contents of onestage are used to update a next stage; and B. means for producing afeedback signal by combining the contents of the stages in accordancewith the following expression: ##EQU10## where the α_(i) 's are thecoefficients of the maximum length polynomial and y_(n-1) is the stateof the (n-i)^(th) stage of the register, and C. coupling means forcoupling an update signal to a first stage of the register, the couplingcircuit adding (1-ε)y_(n) and εS_(n) to produce the update signal y'_(n)=(1-ε)y_(n) +εS_(n), where ε is a coupling factor selected between 0 and1 and S_(n) is a sample of a received signal.
 8. The communicationsystem of claim 6, wherein the coupling factor is set to a valuerelatively close to 1 until the analog shift register synchronizes tothe received signal, and is thereafter decreased.
 9. An analog feedbackshift register for use in a communication system, the shift registerincluding:A. a plurality of stages connected together such that thecontents of one stage are used to update a next stage; and B. means forproducing a feedback signal by combining the contents of the stages inaccordance with the following: ##EQU11## where the α_(i) 's are thecoefficients of the maximum length polynomial and y_(n-1) is the stateof the (n-i)^(th) stage of the register, and C. coupling means forcoupling an update signal y'_(n) to a first stage of the register, thecoupling circuit using as an update y'_(n) =(1-ε)y_(n) +εsgnS_(n), whereε is a coupling factor selected between 0 and 1 and sgnS_(n) is the signof a sample of a received signal.